Problem with bpy.ops.sculpt.brush_stroke
I am unfamiliar with this function in Blender. But suggest possibilities I can! :)

You say, "As far as I know, none of my sculpt settings should depend on the position from which the model is viewed in the 3D viewport (I am using a brush set to Area Plane)" 

That may be, except for one point: the last place that was clicked. Why else have a parameter named "ignore_background_click"??? When you changed viewports, maybe you changed the last place that was clicked? If not, maybe it changes the relative position of that click (if not the absolute position)??

Have you tried setting that parameter to True? In other words you do want to ignore that click position… That's the way I take that parameter name and limited documentation here: 
https://docs.blender.org/api/2.79/bpy.ops.sculpt.html?highlight=bpy%20ops%20sculpt%20brush_stroke#bpy.ops.sculpt.brush_stroke

I expect that you have tried that, so the other possibility is that you could programmatically set that click position (and pass False, as you are).

Why, the possibilities are rampant! :D